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Lhs-equiv

Basic equivalence relation for lhs structures.

Definitions and Theorems

Function: lhs-equiv$inline

(defun lhs-equiv$inline (x y)
  (declare (xargs :guard (and (lhs-p x) (lhs-p y))))
  (equal (lhs-fix x) (lhs-fix y)))

Theorem: lhs-equiv-is-an-equivalence

(defthm lhs-equiv-is-an-equivalence
  (and (booleanp (lhs-equiv x y))
       (lhs-equiv x x)
       (implies (lhs-equiv x y)
                (lhs-equiv y x))
       (implies (and (lhs-equiv x y) (lhs-equiv y z))
                (lhs-equiv x z)))
  :rule-classes (:equivalence))

Theorem: lhs-equiv-implies-equal-lhs-fix-1

(defthm lhs-equiv-implies-equal-lhs-fix-1
  (implies (lhs-equiv x x-equiv)
           (equal (lhs-fix x) (lhs-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: lhs-fix-under-lhs-equiv

(defthm lhs-fix-under-lhs-equiv
  (lhs-equiv (lhs-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-lhs-fix-1-forward-to-lhs-equiv

(defthm equal-of-lhs-fix-1-forward-to-lhs-equiv
  (implies (equal (lhs-fix x) y)
           (lhs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-lhs-fix-2-forward-to-lhs-equiv

(defthm equal-of-lhs-fix-2-forward-to-lhs-equiv
  (implies (equal x (lhs-fix y))
           (lhs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhs-equiv-of-lhs-fix-1-forward

(defthm lhs-equiv-of-lhs-fix-1-forward
  (implies (lhs-equiv (lhs-fix x) y)
           (lhs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhs-equiv-of-lhs-fix-2-forward

(defthm lhs-equiv-of-lhs-fix-2-forward
  (implies (lhs-equiv x (lhs-fix y))
           (lhs-equiv x y))
  :rule-classes :forward-chaining)